Congruent

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(C)Copyright 2020, C. Burke. "AnthroNumerics" is a trademark of Christopher J. Burke and (x, why?).

A Pythagorean Triple whammy.

Originally, I was going to use WALK and RIDE before I decided I wanted Pentagons, for whatever reason. Oddly, it was difficult finding lists of 6-letter words with no repeated letters. Maybe I was using poor search terms.

EDIT: This was supposed to have been added when I got home last night and was sitting at my own computer. That obviously did not happen.
A congurent number is an integer that is the area of a right triagnle with three rational sides, which are not necessarily integers.
In the case of this comic, a triangle with sides of 24/5, 35/12, and 337/60 would have an area of (1/2)(24/5)(35/12), or 7.

No, not an obvious solution. I saw the list of congruent numbers before I found a list of triangles that created them. I wouldn't have wanted to have worked that out. For starters, I probably would've been stuck on the 7-24-25 triangle, trying to get an area of 7 out of that!

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(x, why?) Mini: Happy New Year 2018!

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(C)Copyright 2018, C. Burke.

Happened 24 years ago, and will happen 24 years from now, but it's not a 24-year cycle.

Here's one list I found on the Internet.




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Happy Pythagorean Triple Day!

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(C)Copyright 2017, C. Burke.

This is the first one this century that didn't happen during the school year, unless you count 8/6/10.

I was teaching for 3/4/5, 6/8/10, 5/12/13, and 9/12/15 (and their permutations), although I'd have to check which of those were actually school days and not weekends or holidays.

The ones that we have left to go in this century are 12/16/20, 7/24/25 and 10/24/26. Sadly, I won't be retired by then.




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Problem: Pythagorean Theorem and Tangent-Secant

There are multiple ways to solve problems with circles, but some will be more straightforward than others, based on the information given.

If a circle has a tangent drawn to it, then that tangent is perpendicular to the radius it intersects. In other words, a right angle is created. Generally speaking, that right angle will probably signal the need to use the Pythagorean Theorem at some point in the problem.

On the other hand, if the tangent is accompanied by a secant line, then a second theorem can be plucked from our toolkit: the Tangent-Secant Theorem. If the tangent and secant intersect at a point outside the circle, then the square of the length of the tangent from the external point to the circle will be equal to the product of the portion of the secant outside the circle times the length of the entire segment.

Consider the problem below:

Which of the two theorems do we need to use?

The answer is: either one of them.

The circle has three radii drawn, but only one is labeled. Write the "6" next to the other two segments.

You can now solve for x using the right triangle with legs 6 and x, and with hypotenuse 10. Or you can solve for tangent with length x using the secant with a length of 16 and an external length of 4.

If you choose to work both of them out, you'll find the same answer.

Keeping that in mind, look at this next problem:

Now you will see that you can make a diameter from the given radius, and create a secant. This gives you a second option for some for x.